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| View Original: | Regular polygons meeting at vertex 3 3 7 42.svg (672x774) | |||
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| Keywords: Regular polygons meeting at vertex 3 3 7 42.svg Combinations of regular polygons that can meet at a vertex For Euclidean tilings the internal angles of the polygons meeting at a vertex must add to 360 degrees There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons yielding twenty-one types of vertex Only eleven of these can occur in a uniform tiling of regular polygons In particular if three polygons meet at a vertex and one has an odd number of sides the other two polygons must be the same If they are not they would have to alternate around the first polygon which is impossible if its number of sides is odd These arrangements are here enumerated 3 polygons of sides 3 7 42 3 polygons of sides 3 8 24 3 polygons of sides 3 9 18 3 polygons of sides 3 10 15 3 polygons of sides 3 12 12 3 polygons of sides 4 5 20 3 polygons of sides 4 6 12 3 polygons of sides 4 8 8 3 polygons of sides 5 5 10 3 polygons of sides 6 6 6 4 polygons of sides 3 3 4 12 4 polygons of sides 3 4 3 12 4 polygons of sides 3 3 6 6 4 polygons of sides 3 6 3 6 4 polygons of sides 4 4 4 4 4 polygons of sides 3 4 4 6 4 polygons of sides 3 4 6 4 5 polygons of sides 3 3 3 3 6 5 polygons of sides 3 3 3 4 4 5 polygons of sides 3 3 4 3 4 6 polygons of sides 3 3 3 3 3 3 2013-06-01 14 17 54 own Dllu cc-zero Uploaded with UploadWizard Regular polygons Polygons that can meet at a vertex Equilateral triangles Regular heptagons | ||||
